226 XIII INTEGRATION
(13.21.9) A ^-measurable function h is v-integrable if and only if
This follows from (13.21.8) and (13.9.13).
(13.21.10) Let A be a ^-measurable set in X x Y.
(i) The set M of all x e X such that the section A(x) is not \i-measurable is
^-negligible', the function jth-»/^*(A(.x)) is ^-measurable; and
*(A) = J V
In particular, ifA(x) is ^-negligible except on a ^-negligible set of values ofx,
then A is ^-negligible.
(ii) If A is v-integrable, then the set of all x e X such that A(x) is not
\L-integrable is ^-negligible; the function x\-*/j,(A(x)) (which is defined almost
everywhere with respect to X) h X-integrable; and v(A) = \fi(A(x)) dk(x).
These assertions are particular cases of (13.21.6), (13.21.8) and (13.21.7).
(13.21.11). Letf(resp.g) be a mapping ofX (resp. Y) into [0, +00]. With
the convention of (1 3.11 ) for products, we have
= ( f *
(188.8.131.52) /(xfoOO d%x) dp(y) = f */(x)
By virtue of (13.21.4), we have
jj /W000 d%x) dn(y) ^ f * dl(x) Ff(x)g(y) dn(y).
On the other hand, for each x e X we have
x) J *f(x)g(y) dn(y) = f *( (
g(y)(13.21.7) the functions x^fax.y) d^(y) are A-integrable and we have